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  <summary>
    <para>
      <codeInline>Black-Merton-Scholes</codeInline>
    </para>
  </summary>
  <introduction>
    <para>
      Pricing, greeks, and implied volatility for calls and puts using the constant volatility
      model of Black-Scholes/Merton.
    </para>
  </introduction>
  <section>
    <title>Overview</title>
    <content>
      <para>
        In 1973 Fischer Black and Myron Scholes figured out how to price an option on a stock without
        needing to know the growth rate of the stock. Their formula involves only the current
        stock price, the volatility of the stock, the risk-free interest rate, and the strike and
        expiration of the option contract.
      </para>
      <para>
        Robert Merton came up with the first mathematically correct proof of the formula and he
        and Scholes shared a Nobel Prize in 1997 for their work. Fischer Black passed away much too
        early in 1995. 
      </para>
      <para>
        They came up with a mathematical relationship between the change in the
        option price and the change in the underlying stock price. That ratio is
        called <newTerm>delta</newTerm> and is used to dynamically replicate the
        option payoff using the underlying. The value of the option in their theory
        is determined by the cost of setting up the initial dynamic hedge.
      </para>
      <para>
        Traders don't take off their delta hedge and put on a new delta hedge
        every time they rehedge to the new delta, 
        they just adjust their existing position in the underlying to the new delta.
        This is called <newTerm>gamma</newTerm>. Traders don't like it when
        gamma is large because it means they are going to be very busy.
      </para>
      <para>
        The Black-Merton-Scholes model assumes volatility is constant. It is not.
        The sensitivity of the price to the volatility is called <newTerm>vega</newTerm>.
      </para>
      <para>
        If the market is not moving, the value of an option tends to decrease over time.
        This is called <newTerm>theta</newTerm>. In the BMS theory this is closely
        related to gamma.
      </para>
    </content>
    <sections>
      <section>
        <title>Introduction</title>
        <content>
          <para>
            Paul Samuelson had already won his Nobel prize in 1970 and published work
            that was very close to Black-Scholes and Merton, but he missed the key
            point that it was unnecessary to estimate the drift component of the stock price
            in order to value an option.
            Merton waited until after Black and Scholes published to give a mathematically
            correct proof of their theorem.
            His first proof assumed bond prices had non zero quadratic variation. Nobody
            goes for that these days. He didn't either by 1977 when he published the
            the canonnical proof still used to this day.
          </para>
        </content>
      </section>
      <section>
        <title>Details</title>
        <content>
          <para>
          </para>
        </content>
      </section>
    </sections>
  </section>
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</developerConceptualDocument>

